Computing the Size of Intervals in the Weak Bruhat Order

The weak Bruhat order on Sn is the partial order ≺ so that σ ≺ τ whenever the set of inversions of σ is a subset of the set of inversions of τ . We investigate the time complexity of computing the size of intervals with respect to ≺. Using relationships between two-dimensional posets and the weak Bruhat order, we show that the size of the interval [σ1, σ2] can be computed in polynomial time whenever σ −1 1 σ2 has bounded width (length of its longest decreasing subsequence) or bounded intrinsic width (maximum width of any non-monotone permutation in its block decomposition). Since permutations of intrinsic width 1 are precisely the separable permutations, this greatly extends a result of Wei. Additionally, we show that, for large n, all but a vanishing fraction of permutations σ in Sn give rise to intervals [id, σ] whose sizes can be computed with a sub-exponential time algorithm. The general question of the difficulty of computing the size of arbitrary intervals remains open.